Forcing isomorphism II

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion script script Q such that, in any script Q-generic extension of the universe, there are non-isomorphic models M₁ and M₂ of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if 'c.c.c.' is replaced by other cardinal-preserving adjectives. We also give an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings.